Summary Tradisional | Second Degree Function: Inputs and Outputs

Contextualization

A quadratic function is a mathematical expression of the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. These functions are typically represented by parabolas, which can open either upwards or downwards depending on the sign of 'a'. If 'a' is positive, the parabola opens upwards, while a negative value for 'a' causes it to open downwards. Understanding this concept is crucial as it applies to various real-life scenarios, from physics to economics.

For instance, in physics, the path of a ball thrown into the air follows a parabolic curve, which is a practical application of quadratic functions. In economics, these functions can be utilized to model certain costs and revenues. Therefore, being able to work with and interpret quadratic functions is a vital skill that aids in problem-solving across different fields.

To Remember!

Definition of Quadratic Function

A quadratic function, or second-degree function, is a polynomial function of degree two. Its general form is expressed as f(x) = ax² + bx + c, where a, b, and c are constants with a ≠ 0. The coefficient 'a' is particularly important, as it determines how the parabola is shaped graphically.

The value of 'a' indicates the curvature of the parabola: if 'a' is positive, the parabola opens upwards, suggesting that the function has a minimum point; if 'a' is negative, it opens downwards, indicating a maximum point. This behavior is crucial for understanding how the function responds to different input values (x).

The constant 'b' impacts the slope of the parabola, shifting the vertex along the x-axis, while the constant 'c' signifies the y-axis intersection point (when x = 0). Each constant plays an essential role in shaping and positioning the parabola on the Cartesian plane.

• General form: f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

• Coefficient 'a' indicates the parabola's concavity.

• Constant 'c' represents the intersection of the parabola with the y-axis.

Graph of a Quadratic Function

The graph of a quadratic function showcases a parabola. The direction in which it curves is dictated by the sign of 'a'. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards. This characteristic is fundamental for recognising the function's maximum or minimum points.

The overall shape of the parabola is also affected by coefficients 'b' and 'c'. The coefficient 'b' changes the slope and position of the vertex along the x-axis, while 'c' determines the point where the parabola intersects the y-axis. The interplay between these coefficients informs the unique shape and location of the parabola.

To graph the function, identifying key points such as the vertex, the roots (or zeros), and the intersection with the y-axis is beneficial. These points create a foundational framework for sketching the parabola and grasping its behaviour concerning the variables involved.

• The graph is a parabola.

• Concavity is determined by the sign of 'a'.

• Coefficients 'b' and 'c' influence the slope and position of the parabola.

Vertex of the Parabola

The vertex of a parabola is a pivotal point that represents either the maximum or minimum of the function, depending on whether 'a' is positive or negative. To calculate the vertex, we use the formulas -b/2a for the x-coordinate and f(-b/2a) for the y-coordinate. These derive from the need to identify the point at which the function's derivative is zero, indicating a turning point.

The x-coordinate, -b/2a, marks the position on the x-axis where the parabola reverses direction. By substituting this x value back into the original function, f(x), we can determine the corresponding y-coordinate, signifying the maximum or minimum value of the function. This point is vital for examining the function's behaviour and tackling optimisation challenges.

The vertex can be viewed as the balance point of the function, offering insightful information about the curvature and trend of the parabola. In practical applications, such as in physics and economics, the vertex might represent the peak of a projectile’s flight or the optimal profit point in a cost scenario.

• The vertex signifies the maximum or minimum point of the function.

• Formulas: -b/2a for the x-coordinate and f(-b/2a) for the y-coordinate.

• The vertex highlights the turning point of the parabola.

Roots of the Quadratic Function

The roots of a quadratic function are the x-values for which f(x) equals zero. To find these roots, we apply Bhaskara's formula: x = [-b ± sqrt(b² - 4ac)] / 2a. This allows us to determine where the parabola cuts through the x-axis, which is essential for visually analysing the function.

The discriminant, represented as Δ = b² - 4ac, plays a crucial role in Bhaskara's formula. The value of the discriminant reveals the nature of the roots: if Δ > 0, the function has two distinct real roots; if Δ = 0, there is one repeated real root; and if Δ < 0, the function has no real roots and only complex roots.

Finding the roots of a quadratic function is fundamental for addressing a wide range of mathematical and applied problems. For example, in physics, the roots may indicate the impact points of an object in motion, while in economics, they may signify the equilibrium points of a financial model.

• Roots are the x-values for which f(x) = 0.

• Bhaskara's formula: x = [-b ± sqrt(b² - 4ac)] / 2a.

• The discriminant Δ determines the nature of the roots.

Input and Output Interpretation

The inputs of a quadratic function are the x-values that can be substituted into the function to produce outputs, which are the resulting values of f(x). Understanding how these inputs and outputs connect is crucial for analysing and applying quadratic functions.

To determine the outputs, we substitute various x-values into the function f(x) = ax² + bx + c. Each x-value corresponds to a particular f(x) value, which can be graphed to visualise the parabola. This relationship between inputs and outputs is key to comprehending the function's behaviour over different intervals.

In real-life scenarios, interpreting the inputs and outputs of a quadratic function can yield valuable insights. For example, in a motion question, the inputs may symbolize time, while the outputs reflect the object's position. In economics, the inputs could denote production quantities while the outputs represent related costs or earnings.

• Inputs (x values) are substituted into the function to yield outputs (f(x) values).

• Each x-value results in a corresponding f(x) value.

• Understanding inputs and outputs is vital in various contexts.

Key Terms

• Quadratic Function: A polynomial function of degree two, expressed as f(x) = ax² + bx + c.

• Parabola: The graphical shape created by a quadratic function.

• Coefficient: The constants a, b, and c found in a quadratic function.

• Vertex: The peak or trough point of a parabola.

• Roots: The values of x for which f(x) = 0.

• Bhaskara's Formula: A formula used to find the roots of a quadratic function.

• Graph: The visual representation of a quadratic function.

• Discriminant: The value Δ = b² - 4ac, which determines the nature of the roots.

Important Conclusions

In this lesson, we delved into quadratic functions, beginning with their definition as an expression of the form f(x) = ax² + bx + c. We highlighted the significance of coefficients a, b, and c, with 'a' influencing the parabola's concavity and 'c' denoting the intersection with the y-axis. Grasping these elements is key for performing graphical analysis and applying quadratic functions practically.

We examined the graphical representation, noting how the parabola opens either upwards or downwards based on the sign of 'a'. We also discussed how to locate the vertex, which indicates the function's peak or lowest point, and how to employ Bhaskara's formula to determine the roots or zeros of the function. This knowledge is essential for solving mathematical and applied problems across diverse fields.

Ultimately, we underscored the importance of interpreting inputs (x values) and outputs (f(x) values) in quadratic functions, demonstrating how to calculate and analyse f(x) for various x values. This understanding equips us to apply quadratic functions in real-world contexts, like in physics and economics, aiding in the resolution of practical challenges.

Study Tips

• Go over the examples and exercises tackled in class, concentrating on applying formulas to find vertices and roots.

• Practice constructing graphs of quadratic functions, identifying key points like vertices, roots, and intersections with the y-axis.

• Investigate and resolve practical problems that utilise quadratic functions, such as projectile trajectories in physics or cost analyses in economics, to solidify theoretical understanding with real-world applications.